using StatsBase , DataFrames , DataFrameMacros
using RCall , CategoricalArrays , TexTables
using Distributions , Gadfly , Compose
using CSV , RDatasets , MLJ
using MLJ : schema
include ("pubh.jl" )
@rlibrary readr
@rimport pubh
Data
The Western Collaborative Group Study (WCGS) is a well known prospective cohort study. Male participants aged 39 to 59 from 10 California companies were originally selected to study the relationship between behaviour pattern and the risk of coronary heart disease (CHD).
wcgs = read_rds ("data/wcgs.rds" ) |> rcopy
wcgs |> schema
┌──────────┬───────────────────────────────┬────────────────────────────────────
│ names │ scitypes │ types ⋯
├──────────┼───────────────────────────────┼────────────────────────────────────
│ id │ Count │ Int64 ⋯
│ age │ Count │ Int64 ⋯
│ height │ Continuous │ Float64 ⋯
│ weight │ Continuous │ Float64 ⋯
│ sbp │ Count │ Int64 ⋯
│ dbp │ Count │ Int64 ⋯
│ chol │ Union{Missing, Count} │ Union{Missing, Int64} ⋯
│ beh_pat │ Multiclass{4} │ CategoricalValue{String, UInt32} ⋯
│ ncigs │ Count │ Int64 ⋯
│ dib_pat │ Multiclass{2} │ CategoricalValue{String, UInt32} ⋯
│ chd │ Multiclass{2} │ CategoricalValue{String, UInt32} ⋯
│ type_chd │ Multiclass{4} │ CategoricalValue{String, UInt32} ⋯
│ time │ Count │ Int64 ⋯
│ arcus │ Union{Missing, Multiclass{2}} │ Union{Missing, CategoricalValue{S ⋯
│ smoker │ Multiclass{2} │ CategoricalValue{String, UInt32} ⋯
└──────────┴───────────────────────────────┴────────────────────────────────────
1 column omitted
The kfm data frame was collected by Kim Fleischer Michaelsen and contains data for 50 infants of age approximately 2 months. They were weighed immediately before and after each breast feeding. and the measured intake of breast milk was registered along with various other data.
kfm = read_rds ("data/kfm.rds" ) |> rcopy
kfm |> schema
┌────────────┬───────────────┬──────────────────────────────────┐
│ names │ scitypes │ types │
├────────────┼───────────────┼──────────────────────────────────┤
│ no │ Count │ Int64 │
│ dl_milk │ Continuous │ Float64 │
│ sex │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ weight │ Continuous │ Float64 │
│ ml_suppl │ Count │ Int64 │
│ mat_weight │ Count │ Int64 │
│ mat_height │ Count │ Int64 │
└────────────┴───────────────┴──────────────────────────────────┘
Data on reported cases of influenza by age group in the 1957 pandemic in England and Wales.
flu = DataFrame (CSV.File ("data/fluraw.csv" ))
flu |> schema
┌───────┬────────────────┬───────┐
│ names │ scitypes │ types │
├───────┼────────────────┼───────┤
│ week │ ScientificDate │ Date │
│ child │ Count │ Int64 │
│ young │ Count │ Int64 │
│ mid │ Count │ Int64 │
│ old │ Count │ Int64 │
└───────┴────────────────┴───────┘
Data on birth weights from newborns in a London Hospital.
birth = read_rds ("data/birthwt.rds" ) |> rcopy
birth |> schema
┌───────┬───────────────┬──────────────────────────────────┐
│ names │ scitypes │ types │
├───────┼───────────────┼──────────────────────────────────┤
│ low │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ age │ Count │ Int64 │
│ lwt │ Continuous │ Float64 │
│ race │ Multiclass{3} │ CategoricalValue{String, UInt32} │
│ smoke │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ ptl │ Count │ Int64 │
│ ht │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ ui │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ ftv │ Count │ Int64 │
│ bwt │ Count │ Int64 │
│ Race │ Multiclass{2} │ CategoricalValue{String, UInt32} │
└───────┴───────────────┴──────────────────────────────────┘
Data on the energy expenditure in groups of lean and obese women.
energy = rcopy (R"ISwR::energy" )
energy |> schema
┌─────────┬───────────────┬──────────────────────────────────┐
│ names │ scitypes │ types │
├─────────┼───────────────┼──────────────────────────────────┤
│ expend │ Continuous │ Float64 │
│ stature │ Multiclass{2} │ CategoricalValue{String, UInt32} │
└─────────┴───────────────┴──────────────────────────────────┘
Fisher’s iris dataset on measurements for three species of iris.
iris = rcopy (R"datasets::iris" )
iris |> schema
┌──────────────┬───────────────┬──────────────────────────────────┐
│ names │ scitypes │ types │
├──────────────┼───────────────┼──────────────────────────────────┤
│ Sepal_Length │ Continuous │ Float64 │
│ Sepal_Width │ Continuous │ Float64 │
│ Petal_Length │ Continuous │ Float64 │
│ Petal_Width │ Continuous │ Float64 │
│ Species │ Multiclass{3} │ CategoricalValue{String, UInt32} │
└──────────────┴───────────────┴──────────────────────────────────┘
Distributions
Histograms
We use histograms to look at the distribution of continuous variables. Histograms make most sense for relatively large data (\(\geq 100\) observations). In this fist histogram, we show the density.
plot (
birth,
x = : bwt, Guide.xlabel ("Birth weight (g)" ),
Guide.ylabel ("Density" ),
Geom.histogram (bincount= 30 , density= true )
)
The default for histograms is to show the absolute frequency (counts):
plot (
wcgs,
x = : sbp,
Geom.histogram (bincount= 30 ),
Guide.xlabel ("SBP (mm Hg)" ),
Guide.ylabel ("Counts" ),
Coord.cartesian (xmin= 80 , ymax= 700 )
)
We can compare distributions using the density or the probability density function (pdf), as shown here:
plot (
wcgs,
x = : sbp, xgroup = : chd,
Geom.subplot_grid (
Geom.histogram (bincount= 30 , density= true )
),
Guide.xlabel ("SBP (mm Hg)" ),
Guide.ylabel ("Density" )
)
Density Plots
Another way to look at the distribution of a continuous variable is with density plots.
plot (
wcgs,
x = : sbp, color = : chd,
Geom.density,
Guide.xlabel ("SBP (mm Hg)" ),
Guide.ylabel ("Density" ),
Guide.colorkey (title= "CHD" ),
style (line_width= 1 mm),
Coord.cartesian (xmin= 80 )
)
Diastolic blood pressure (DBP) is skewed to the right (see the QQ-Plot). In the following example, we log-transform DBP.
plot (
wcgs,
x = log .(wcgs.dbp), color = : chd,
Geom.density,
Guide.xlabel ("log (DBP)" ),
Guide.ylabel ("Density" ),
Guide.colorkey (title= "CHD" ),
style (line_width= 1 mm),
)
We have missing values in chol, thus we need to drop them before constructing a density plot or else, we would have an error.
plot (
@subset (wcgs, !ismissing (: chol)),
x = : chol, color = : chd,
Geom.density,
Guide.xlabel ("Cholesterol (mg/dl)" ),
Guide.ylabel ("Density" ),
Guide.colorkey (title= "CHD" ),
style (line_width= 1 mm),
Coord.cartesian (ymax= 0.011 )
)
QQ-Plots
The best way to determine if a continuous variable is normally distributed or not is with quantile-quantile plots (QQ-plots). We plot the quantiles of our variable of interest against quantiles from the standard normal distribution (which has a mean \(\mu=0\) and a standard deviation \(\sigma=1\) ). This type of QQ-plots against the normal distribution are known as QQ-normal plots. If the variable is normally distributed, then a linear relationship will be observed.
qq_plot (birth.bwt; ylab = "Birth weight (g)" )
The function qq_plot is part of pubh.jl, a wrapper to functions from Gadfly constructs QQ-plots.
Right-skewed variables, show an upper right curve in the QQ-plot:
qq_plot (wcgs.dbp; ylab= "DBP (mm Hg)" )
The next plot, shows the use of log scales.
plot (
wcgs,
x= fit .([Normal], [wcgs.dbp]), y=: dbp,
Stat.qq,
Guide.xlabel ("Theoretical quantiles" ),
Guide.ylabel ("DBP (mm Hg)" ),
Scale.y_log10,
Theme (default_color= "IndianRed" )
)
We can plot the log of the variable, instead of using log-scales.
qq_plot (log .(wcgs.dbp), ylab= "log (DBP)" )
Theoretical quantiles
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4
5
log (DBP)
Associations Between Continuous Variables
Scatter Plots
We use scatter plots to look at the relationship between two continuous variables. By default, the dependent variable (response) is plotted on the \(y\) -axis while the independent variable (explanatory) is plotted on the \(x\) -axis.
From the kfm dataset, let’s see if there is a relationship between the weight of the mother and the breast-milk intake of the child.
plot (
kfm,
x = : mat_weight, y = : dl_milk, color = : sex,
Guide.xlabel ("Maternal weight (kg)" ),
Guide.ylabel ("Breast milk intake (dl/day) " )
)
We can extend plots by adding new layers . In the following example, we use two layers: one for the scatter plot and a second to add a line representing a linear fit.
plot (
kfm,
x = : mat_weight, y = : dl_milk, color = : sex,
Geom.point,
layer (
Stat.smooth (method=: lm),
Geom.line,
Geom.ribbon
),
Guide.xlabel ("Maternal weight (kg)" ),
Guide.ylabel ("Breast milk intake (dl/day) " )
)
In the next plot, instead of showing the linear fit, we add a loess trend with a smoother:
plot (
kfm,
x = : mat_weight, y = : dl_milk, xgroup = : sex,
Geom.subplot_grid (Geom.point, Geom.smooth (smoothing= 0.9 )),
Guide.xlabel ("Maternal weight (kg)" ),
Guide.ylabel ("Breast milk intake (dl/day) " )
)
In the next plot, we use different line colours and different symbols (by sex):
plot (
kfm,
x = : mat_weight, y = : dl_milk,
color = : sex, shape = : sex,
Geom.point, Geom.smooth (method = : lm),
Guide.xlabel ("Maternal weight (kg)" ),
Guide.ylabel ("Breast milk intake (dl/day) " )
)
Line Charts
In this example, our dataset is in wide format.
Reshaping wide to long data:
flu_melt = stack (flu, Not (: week))
flu_melt |> head
1
1957-08-25
child
0
2
1957-09-01
child
0
3
1957-09-08
child
0
4
1957-09-15
child
23
5
1957-09-22
child
63
plot (
flu_melt,
x = : week, y = : value, color = : variable,
Geom.point, Geom.path,
Guide.xlabel ("Date" ),
Guide.ylabel ("Number of cases" ),
Guide.colorkey ("Age group" )
)
Comparing Groups
Box-Plots
When we are comparing continuous variables, between two or more groups, box plots are the best option, particularly if the number of observations in the groups is relatively large \((\geq 30)\) .
plot (
@subset (wcgs, !ismissing (: chol)),
x = : chd, y = : chol,
Geom.boxplot,
Guide.xlabel ("Coronary Heart Disease" ),
Guide.ylabel ("Cholesterol (mg/dl)" )
)
In the previous figure the presence of an outlier is clear. If we would like to remove that outlier, we would need to declare that in the report. For demonstration purposes, if we do not want to show the outlier in the plot, we have the option to filter the data using the @subset command.
plot (
@subset (wcgs |> dropmissing, : chol .< 500 ),
x = : chd, y = : chol,
Geom.boxplot,
Guide.xlabel ("Coronary Heart Disease" ),
Guide.ylabel ("Cholesterol (mg/dl)" ),
Theme (default_color= "MidnightBlue" )
)
Coronary Heart Disease
No CHD
CHD
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Cholesterol (mg/dl)
plot (
birth,
x = : race, y = : bwt, color = : smoke,
Geom.boxplot,
Guide.xlabel ("" ),
Guide.ylabel ("Birth weight (g)" ),
Guide.colorkey ("" )
)
Faceting:
plot (
birth,
x = : Race, y = : bwt, color=: Race, xgroup = : smoke,
Geom.subplot_grid (Geom.boxplot),
Guide.xlabel ("" ),
Guide.ylabel ("Birth weight (g)" ),
Guide.colorkey ("" )
)
Strip Charts
When we want to compare groups and the number of observations is relatively small \((n<30)\) , strip charts are superior than box-plots as we can show all observations.
strip_error (
energy, "stature" , "expend" ,
xlab = "Stature" ,
ylab = "Energy expenditure (MJ)"
)
Error Bars
birth_bst = combine (groupby (birth, [: smoke, : race]), : bwt => cis => AsTable)
1
Non-smoker
White
3428.75
3218.93
3638.57
2
Non-smoker
African-American
2854.5
2550.09
3158.91
3
Non-smoker
Other
2815.78
2628.31
3003.25
4
Smoker
White
2826.85
2656.57
2997.12
5
Smoker
African-American
2504.0
2109.15
2898.85
6
Smoker
Other
2757.17
2298.84
3215.49
plot (
birth_bst,
x=: race, y=: outcome, xgroup=: smoke,
ymin=: lower, ymax=: upper,
Geom.subplot_grid (
Geom.point,
Geom.errorbar
),
Guide.ylabel ("Birth weight (g)" ),
Guide.xlabel ("" ),
Theme (default_color= "Firebrick" )
)
Violin Plots
Violin plots are an alternative to box-plots.
plot (
kfm,
x = : sex, y = : dl_milk,
Geom.violin,
Guide.xlabel ("" ),
Guide.ylabel ("Breast milk intake (dl/day) " ),
Theme (default_color= "Plum" )
)
plot (
wcgs,
x = : beh_pat, y = : sbp, color = : beh_pat, xgroup=: chd,
Geom.subplot_grid (Geom.violin),
Guide.xlabel ("Coronary Heart Disease status" ),
Guide.ylabel ("SBP (mmHg)" ),
Guide.colorkey ("Behaviour \n Pattern" )
)
Bar Charts
Bar charts or bar plots are used, most of the time, to display information from a table graphically. Therefore, for publications are not the best option as a table uses less space and contains better information.
Frequency
For categorical variables, we can plot the counts with frequency.
@combine (groupby (birth, [: race, : low]), : count = @nrow )
1
White
No
73
2
White
Yes
23
3
African-American
No
15
4
African-American
Yes
11
5
Other
No
42
6
Other
Yes
25
plot (
@combine (groupby (birth, [: race, : low]), : count = @nrow ),
x = : race, y = : count, color = : low,
Geom.bar (position=: dodge),
Guide.xlabel ("" ),
Guide.colorkey (title= "Low birth weight?" , pos= [0.4 w, - 0.3 h])
)
Expectation
For continuous variables, we can plot the expected (mean) values, with expectation.
combine (groupby (kfm, : sex), : dl_milk => cis => AsTable)
1
Boy
7.9524
7.36768
8.53712
2
Girl
7.0564
6.49906
7.61374
plot (
combine (groupby (kfm, : sex), : dl_milk => cis => AsTable),
x = : sex, y = : outcome, ymin = : lower, ymax = : upper,
Scale.x_discrete (levels= ["Boy" , "Girl" ]),
Geom.errorbar, Stat.dodge, Geom.bar (position = : dodge),
Guide.xlabel ("" ),
Guide.ylabel ("Breast-milk intake (dl/day)" )
)
combine (groupby (birth, [: smoke, : Race]), : bwt => cis => AsTable)
1
Non-smoker
White
3428.75
3218.93
3638.57
2
Non-smoker
Non-white
2824.51
2664.85
2984.17
3
Smoker
White
2826.85
2656.57
2997.12
4
Smoker
Non-white
2642.09
2336.65
2947.54
plot (
combine (groupby (birth, [: smoke, : Race]), : bwt => cis => AsTable),
x = : Race, y = : outcome, color = : smoke,
ymin = : lower, ymax = : upper,
Scale.x_discrete (levels= ["White" , "Non-white" ]),
Geom.errorbar, Stat.dodge,
Geom.bar (position = : dodge),
Guide.xlabel ("" ),
Guide.ylabel ("Birth weight (g)" ),
Guide.colorkey (title= "" , pos= [0.8 w, - 0.4 h])
)